Triangle \(ABC\) has \(BC=20.\) The incircle of the triangle evenly trisects the median \(AD.\) If the area of the triangle is \(m \sqrt{n}\) where \(m\) and \(n\) are integers and \(n\) is not divisible by the square of a prime, find \(m+n.\)

(第二十三届AIME1 2005 第15题)